You remember set theory, the thing you learnt in 8th grade in school. The thing where you take set of all horses, the set of all things that are brown, and do some operations on them like unions or differences and somehow prove that not all horses are brown.
Set theory is the foundation of mathematics. Working from it you can spawn all other branches of mathematics. The same is not true for any of the other mathematical branches, which means set theory is somehow special and more important than the rest. So any change here is bound to send big ripples into all directions, not least into the realm of theoretical physics.
First let me set the stage:
Since the universe and existence of things within the universe appear to be founded on physics, which is founded on mathematics, which is founded on set theory… you should suppose that everything you see around you, in the very place you are right now, are sets and nothing but sets. Even the actions they perform must somehow also be sets. You are watching sets have set-actions done to them.
Now let’s turn to the problem. The standard naive set theory has a fundamental problem nicely illustrated by Russell’s Paradox. Briefly stated, naive set theory has so few limitations on what is legal to do, that you can conjure up statements within it that are self-contradictory. In other words: utter nonsense. As a result, naive set theory contradicts itself and cannot possibly be true. Russels’s Paradox is the fact that within naive set theory you can say:
“Let S be the set of all sets that do not contain themselves”
…and have it be a legal thing to say. But examining what that logically means (does S contain itself or not?) quickly leads to a self-contradiction: such a set is clearly impossible! But the problem is even deeper than that. Not only was the set itself illegal, but the very act of saying that sentence was illegal. It was illegal in the same way as saying “let 3 be equal to 5” is illegal. There was nothing wrong with the 3 or the 5. It was the sentence itself that was wrong. There clearly must be something wrong with saying such a thing! But what?! What exactly is the problem of saying that sentence?
Since set theory have been shown to be the foundational theory for all other mathematical theories, fixing this bug has attracted very many of the worlds brightest minds for over a hundred years!
There have historically been two kinds of approaches to try to “fix” this bug of naive set theory. One is axiomatic set theory (of which Zermelo-Fränkel set theory is the most well known) and the other is type theory. Type theory attempts to fix the problem by introducing “set types”, much like how variables in computer programs have types. Having each operation spawn a new result of a new type, and limiting what kinds of types are allowed to have certain things done to them, is a neat way to get rid of the paradoxes. Just think a lot and add the right types and the right rules and you will manage.
Axiomatic set theory works the other way; by taking things away. Naive set theory has a whole bunch of concepts that seem fine, and actions that at first appear like they should be legal things to do. Axiomatic set theory supposes that, since these apparently legal things lead to contradictions, at least some of them must actually be illegal despite how they appear! So it tries to get to the roots of set theory itself and as clearly as possible list the facts (axioms) about sets that we assume to be true.
I claim that the type theory approach has a fundamental problem. I also claim that the axiomatic set theory approach can be stated in another way that should appear more intuitive to people, which hints at a direction in which axiomatic set theory should be further developed.
Turtles and the origin of the universe
A well-known scientist (some say it was Bertrand Russell) once gave a public lecture on astronomy. He described how the earth orbits around the sun and how the sun, in turn, orbits around the center of a vast collection of stars called our galaxy. At the end of the lecture, a little old lady at the back of the room got up and said: “What you have told us is rubbish. The world is really a flat plate supported on the back of a giant tortoise.” The scientist gave a superior smile before replying, “What is the tortoise standing on?” “You’re very clever, young man, very clever,” said the old lady. “But it’s turtles all the way down!”
— Stephen Hawking, 1988[1]
If you ask yourself “Where did all things come from? What is the origin of the universe?” what kind of answer pops up for you? Some people answer that God created it. But who, then, created god? Another god? Are there gods all the way up?
Some people answer that the Big Bang created everything. But what then created the big bang? Are there bangs all the way inward? What would that even look like?
We can sit and conjure up wild ideas all day long but in the end it will mean nothing. We need to look at the evidence the world presents to us, examine them carefully, and come to conclusions. And all the evidence points to a Big Bang kind of event. But how could that possibly work out? How can something come from nothing?
The current answer is: We do not yet know. But if we follow our intuition, nothing is less than little which is less than more. So if we are to go from today, where there are many things, towards the Big Bang, where there is nothing, we need to have less and less things the closer we get to the Big Bang. Or else nothing makes sense. The universe could, of course, be completely nonsensical, but from a human point of view… that would totally suck. So let’s assume that the universe does, taken as a whole, make sense.
So if we are to imagine a way for the Big Bang to at all be possible, there somehow needs to be set theory there, present at the very beginning. Set theory must somehow have existed before the Big Bang happened. So if we go backwards in time from there, when we get closer and closer to the Big Bang, as things start disappearing one after another, set theory must be one of the very last things to disappear! This hints at a possibility: Let us assume that set theory somehow caused the Big Bang.
But what, then, created set theory? Let us suppose that there was another kind of bang, but a more invisible and abstract one, which first created set theory. And have that bang be of a special type that indeed did come from nothing. Is this chain of events plausible? Or even possible?
The principle that there are less and less things the closer we get to the Big Bang indicates that Type Theory (the theory in which things were added to Set Theory) is not what the world is based upon, since if you gradually remove elements from such a theory it disintegrates into a state where it is self-contradictory again. It would not survive a walk backwards in time towards the Big Bang. While it is mathematically sound right now, it wasn’t always so. Viewed in that light, it appears that the world is most likely based upon Axiomatic Set Theory (the theory in which things were removed from Set Theory). Since on its journey from its present state towards the state where it doesn’t exist at all, when having more and more elements removed from it, it doesn’t pass through any state that is self-contradicting. So intuition says that Axiomatic Set Theory is the path to follow and that it may indeed be the actual chain of events of the entire universe.
Let’s examine this over the next series of articles!
I have a line of arguments which describe a new kind of Axiomatic Set Theory that illustrates that this is entirely possible.